22 July 2021
Original Citation:
Applying a description logic of typicality as a generative tool for concept combination in computational creativity
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DOI:10.3233/IA-180016
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Applying a Description Logic of Typicality as
a Generative Tool for Concept Combination
in Computational Creativity
Antonio Lieto
a,b,∗Gian Luca Pozzato
aaDipartimento di Informatica, Università di Torino, c.so Svizzera 185, 10149 Torino, Italy
E-mail: {antonio.lieto,gianluca.pozzato}@unito.it
bICAR-CNR, via Ugo La Malfa 153, 90146 Palermo, Italy
Abstract. In this work we exploit a nonmonotonic Description Logic of typicality as a tool for the generation and the exploration of novel creative concepts. Our logic, called TCL, is able to deal with the phenomenon of prototypical concept combination, which has been shown to be problematic to be modelled in formalisms like fuzzy logic. The logic TCLis the result of combining three main ingredients: (i) the logic of typicality ALC + TR, that allows one to describe typical properties of a class by means of a typicality operator, (ii) the distributed semantics of probabilistic Description Logics, that allows one to provide a probability distribution over worlds/scenarios, and (iii) established heuristics used by humans for concept composition coming from the field of cognitive semantics. Our approach could be useful in many applicative scenarios, ranging from video games to the creation of new movie characters.
Keywords: Description Logics, Nonmonotonic Reasoning, Typicality, Computational Creativity, Concept Generation
1. Introduction
The automatic generation of novel concepts, ob-tained by coupling the typical knowledge of pre-existing ones, is an important trait of human creativity. Dealing with this problem requires, from an AI per-spective, the harmonization of two conflicting require-ments that are hardly accommodated in symbolic sys-tems: the need of a syntactic compositionality (typical of logical systems) and one concerning the exhibition of typicality effects [15]. According to a well-known argument [23], in fact, prototypical concepts are not compositional. The argument runs as follows: consider a concept like pet fish. It results from the composition of the concept pet and of the concept fish. However, the prototype of pet fish cannot result from the compo-sition of the prototypes of a pet and a fish: e.g. a typi-cal pet is furry and warm, a typitypi-cal fish is grayish, but
*Corresponding author. E-mail: antonio.lieto@unito.it.
a typical pet fish is neither furry and warm nor grayish (typically, it is red).
In this work we provide a framework able to ac-count for this type of human-like concept combination in the scenario of typical concept invention. We exploit a nonmonotonic Description Logic (from now on DL) of typicality called TCL(typical compositional logic),
that has been already shown able to capture well estab-lished examples in the literature of cognitive science concerning concept combination, in particular we have shown that the logic TCL is able to model the above
mentioned pet fish composition (i.e. the guppy effect), the conjunction fallacy problem (also known as the Linda problem) and the phenomena of metaphorical concept composition. We remind the interested reader to [18,19].
This logic combines three main ingredients. The first one relies on the DL of typicality ALC + TR
in-troduced in [7,8]. In this logic, “typical” properties can be directly specified by means of a “typicality”
ator T enriching the underlying DL, and a TBox can contain inclusions of the form T(C) v D to repre-sent that “typical Cs are also Ds”1. As a difference
with standard DLs, in the logic ALC + TR one can
consistently express exceptions and reason about de-feasible inheritance as well. For instance, a knowledge base can consistently express that “normally, athletes are fit”, whereas “sumo wrestlers usually are not fit” by T(Athlete) v Fit and T(SumoWrestler ) v ¬Fit , given that SumoWrestler v Athlete. The semantics of the T operator is characterized by the properties of rational logic[13], recognized as the core properties of nonmonotonic reasoning. ALC + TRis characterized
by a minimal model semantics corresponding to an ex-tension to DLs of a notion of rational closure as de-fined in [13] for propositional logic: the idea is to adopt a preference relation among ALC +TRmodels, where
intuitively a model is preferred to another one if it con-tains less exceptional elements, as well as a notion of minimal entailmentrestricted to models that are mini-mal with respect to such preference relation. As a con-sequence, T inherits well-established properties like specificityand irrelevance: in the example, the logic ALC + TRallows us to infer that, normally, an athlete
who is also bald is fit, i.e. T(Athlete u Bald ) v Fit (being bald is irrelevant with respect to being fit) and, if one knows that Hiroyuki is a typical sumo wrestler, to infer that he is not fit, giving preference to the most specific information.
As a second ingredient, we consider a distributed se-mantics similar to the one of probabilistic DLs known as DISPONTE [27], allowing one to label axioms with degrees representing probabilities, but restricted to typicality inclusions. The basic idea is to label in-clusions T(C) v D with a real number between 0.5 and 1, representing its probability2: such a number rep-resents the probability of finding elements of C being also D, then we impose that it is at least 50% since we are only considering typicality properties. We as-sume that the axioms are independent from each other.
1In this paper we use the term properties as intended in classical
logic. In this respect, in Description Logics, for an inclusion relation of the form either C v D or T(C) v D, we say that D is a property of the concept C (as we will see later, either a rigid or a typical one).
2Probabilities for typicality inclusions are assumed to come from
an application domain as in any probabilistic formal framework (probabilities in probabilistic extensions of logic programs, degrees of belief in fuzzy logics). Here, we focus on the proposal of the formalism itself, therefore the machinery for obtaining probabilities from an application domain will not be discussed.
The resulting knowledge base defines a probability distribution over scenarios: roughly speaking, a sce-nario is obtained by choosing, for each typicality in-clusion, whether it is considered as true of false. In a slight extension of the above example, we could have the need of representing that both typicality inclusions about athletes and sumo wrestlers have a probability of 80%, whereas we also believe that athletes are usually young with a higher probability of 95%, with the fol-lowing KB: (1) SumoWrestler v Athlete; (2) 0.8 :: T(Athlete) v Fit ; (3) 0.8 :: T(SumoWrestler ) v ¬Fit ; (4) 0.95 :: T(Athlete) v YoungPerson. We consider eight different scenarios, representing all pos-sible combinations of typicality inclusion: as an ex-ample, {((2), 1), ((3), 0), ((4), 1)} represents the sce-nario in which (2) and (4) hold, whereas (3) does not. We equip each scenario with a probability depending on those of the involved inclusions, then we restrict reasoning to scenarios whose probabilities belong to a given and fixed range.
As a third element of the proposed formalization we employ a method inspired by cognitive semantics [10] for the identification of a dominance effect between the concepts to be combined: for every combination, we distinguish a HEAD, representing the stronger element of the combination, and a MODIFIER. The basic idea is: given a KB and two concepts CH(HEAD) and CM
(MODIFIER) occurring in it, we consider only some scenarios in order to define a revised knowledge base, enriched by typical properties of the combined concept C v CH u CM (the heuristics for the scenario
selec-tions are detailed in Section 2, Definition 7).
In this work, we show that the proposed logic TCL
is able to tackle the problem of composing prototyp-ical concepts (by creating a novel, plausible, concep-tual prototype) and we exploit the proposed formalism as a tool for the generation of novel creative concepts, that could be useful in many applicative scenarios. We also outline that the reasoning complexity of TCL is
EXPTIME-complete, as in the standard ALC, witness-ing that the proposed approach is promiswitness-ing since it re-mains in the same complexity class of the underlying logic.
The plan of the paper is as follows. In Section 2 we first describe the logic TCL presented in [18], then in
Section 3 we introduce COCOS, a tool implementing reasoning service in such a logic. Section 4 describes some applications of the proposed approach. We con-clude in Section 5 with a discussion on related ap-proaches and some pointers to future works. This
pa-per extends and revises preliminary results presented in [17].
2. A Logic for Concept Combination
The nonmonotonic Description Logic TCL
com-bines the semantics based on the rational closure of ALC + TR[7] with the probabilistic DISPONTE
se-mantics [27]. By taking inspiration from [21], we sider two types of properties associated to a given con-cept: rigid and typical. Rigid properties are those that hold under any circumstance, e.g. C v D (all Cs are Ds). Typical properties are represented by inclusions equipped by a probability. Additionally, we employ a cognitive heuristic for the identification of a domi-nance effect between the concepts to be combined, dis-tinguishing between HEAD and MODIFIER3.
The language of TCLextends the basic DL ALC by
typicality inclusionsof the form T(C) v D equipped by a real number p ∈ (0.5, 1) representing its proba-bility, whose meaning is that “normally, Cs are also D with probability p”.
Definition 1 (Language of TCL) We consider an al-phabet C of concept names, R of role names, and O of individual constants. GivenA ∈ C and R ∈ R, we de-fine:
C, D := A | > | ⊥ | ¬C | C u C | C t C | ∀R.C | ∃R.C We define a knowledge baseK = hR, T , Ai where:
– R is a finite set of rigid properties of the form C v D;
– T is a finite set of typicality properties of the form p :: T(C) v D, where p ∈ (0.5, 1) ⊆ R is the probability of the inclusion;
– A is the ABox, i.e. a finite set of formulas of the form eitherC(a) or R(a, b), where a, b ∈ O. It is worth noticing that we avoid typicality inclusions with degree 1. Indeed, an inclusion 1 :: T(C) v D would mean that it is a rigid property, that we repre-sent with C v D ∈ R. Also, observe that we only allow typicality inclusions equipped with probabilities p > 0.5. Indeed, the very notion of typicality derives from the one of probability distribution, in particular typical properties attributed to entities are those
char-3Here we assume that some methods for the automatic assignment
of the HEAD/MODIFER pairs are/may be available and focus on the discussion of the reasoning part.
acterizing the majority of instances involved. More-over, in our effort of integrating two different seman-tics – DISPONTE and typicality logic – the choice of having probabilities higher than 0.5 for typicality in-clusions seems to be the only compliant with both for-malisms. In fact, despite the DISPONTE semantics al-lows one to assign also low probabilities/degrees of be-lief to standard inclusions, in the logic TCL it would
be misleading to also allow low probabilities for typ-icality inclusions. Please, note that this is not a limi-tation of the expressivity of the logic TCL: we can in
fact represent properties not holding for typical mem-bers of a category, for instance if one need to represent that typical students are not married, we can have that 0.8 :: T(Student ) v ¬Married .
Following from the DISPONTE semantics, each ax-iom is independent from each others. This allows us to deal with conflicting typical properties equipped with different probabilities.
A model M of TCLextends standard ALC models
by a preference relation among domain elements as in the logic of typicality [7]. In this respect, x < y means that x is “more normal” than y, and that the typical members of a concept C are the minimal elements of C with respect to this relation. An element x ∈ ∆Iis a typical instance of some concept C if x ∈ CI and there is no element in CImore typicalthan x. Definition 2 (Model) A model M is any structure
h∆I, <, .Ii where:
– ∆Iis a non empty set of items called the domain; – < is an irreflexive, transitive, well-founded and modular (for allx, y, z in ∆I, ifx < y then either x < z or z < y) relation over ∆I;
– .I is the extension function that maps each con-ceptC to CI ⊆ ∆I, and each roleR to RI ⊆
∆I× ∆I. For concepts ofALC, CIis defined as
usual. For theT operator, we have (T(C))I = M in<(CI),
where M in<(CI) = {x ∈ CI | @y ∈
CIs.t.y < x}.
A model M can be equivalently defined by postulating the existence of a function kM : ∆I 7−→ N, where
kMassigns a finite rank to each domain element [7]:
· · · < x from x to a minimal x0, i.e. such that there is
no x0such that x0 < x0. The rank function kMand <
can be defined from each other by letting x < y if and only if kM(x) < kM(y).
Definition 3 (Model satisfying a KB) Let K = hR, T , Ai be a KB. Given a modelM = h∆I, <, .Ii, we assume
that.I is extended to assign a domain elementaI of ∆Ito each individual constanta of O. We say that:
– M satisfies R if, for all C v D ∈ R, we have CI ⊆ DI;
– M satisfies T if, for all p :: T(C) v D ∈ T , we have thatT(C)I⊆ DI, i.e.M in
<(CI) ⊆ DI;
– M satisfies A if, for each assertion F ∈ A, if F = C(a) then aI ∈ CI, otherwise if F = R(a, b) then (aI, bI) ∈ RI.
Even if the typicality operator T itself is nonmono-tonic (i.e. T(C) v E does not imply T(C u D) v E), what is inferred from a KB can still be inferred from any KB’ with KB ⊆ KB’, i.e. the resulting logic is monotonic. In order to perform useful nonmonotonic inferences, in [7] the authors have strengthened the above semantics by restricting entailment to a class of minimal models. Intuitively, the idea is to restrict entailment to models that minimize the atypical in-stances of a concept. The resulting logic corresponds to a notion of rational closure on top of ALC + TR.
Such a notion is a natural extension of the rational clo-sure construction provided in [13] for the propositional logic. This nonmonotonic semantics relies on minimal rational models that minimize the rank of domain el-ements. Informally, given two models of KB, one in which a given domain element x has rank 2 (because for instance z < y < x), and another in which it has rank 1 (because only y < x), we prefer the latter, as in this model the element x is assumed to be “more typical” than in the former. Query entailment is then restricted to minimal canonical models. The intuition is that a canonical model contains all the individuals that enjoy properties that are consistent with KB. This is needed when reasoning about the rank of the con-cepts: it is important to have them all represented. A query F is minimally entailed from a KB if it holds in all minimal canonical models of KB. In [7] it is shown that query entailment in the nonmonotonic ALC + TR
is in EXPTIME.
Definition 4 (Entailment) Let K = hR, T , Ai be a KB and letF be either C v D (C could be T(C0)) or
C(a) or R(a, b). We say that F follows from K if, for all minimalM satisfying K, then also M satisfies F .
Let us now define the notion of scenario of the com-position of concepts. Intuitively, a scenario is a knowl-edge base obtained by adding to all rigid properties in R and to all ABox facts in A only some typicality properties. More in detail, we define an atomic choice on each typicality inclusion, then we define a selection as a set of atomic choices in order to select which typ-icality inclusions have to be considered in a scenario. Definition 5 (Atomic choice) Given K = hR, T , Ai, whereT = {E1 = p1 :: T(C1) v D1, . . . , En =
pn :: T(Cn) v Dn} we define (Ei,ki) anatomic
choice for some i ∈ {1, 2, . . . , n}, where ki∈ {0, 1}.
Definition 6 (Selection) Given K = hR, T , Ai, where T = {E1 = p1 :: T(C1) v D1, . . . , En = pn ::
T(Cn) v Dn} and a set of atomic choices ν, we say
that ν is a selection if, for each Ei, one decision is
taken, i.e. either (Ei, 0)∈ ν and (Ei, 1)6∈ ν or (Ei, 1)
∈ ν and (Ei, 0)6∈ ν for i = 1, 2, . . . , n. The
probabil-ity ofν is P (ν) = Q
(Ei,1)∈ν
pi Q
(Ei,0)∈ν
(1 − pi).
Definition 7 (Scenario) Given K = hR, T , Ai, where T = {E1 = q1 :: T(C1) v D1, . . . , En = qn ::
T(Cn) v Dn} and given a selection σ, we define a
scenario wσ = hR, {Ei | (Ei, 1) ∈ σ}, Ai. We also
define the probability of a scenariowσas the
probabil-ity of the corresponding selection, i.e.P (wσ) = P (σ).
Last, we say that a scenario isconsistent when it ad-mits a model in the logicTCL.
2.1. Selected scenarios
We denote with WK the set of all scenarios for a
knowledge base K. It immediately follows that the probability of a scenario P (wσ) is a probability
distri-bution over scenarios, that is to say P
w∈WK
P (w) = 1. Given a KB K = hR, T , Ai and given two con-cepts CH and CM occurring in K, our logic allows
one to define the compound concept C as the combina-tion of the HEAD CHand the MODIFIER CM, where
C v CHu CM and the typical properties of the form
T(C) v D to ascribe to the concept C are obtained in the set of scenarios that:
1. are consistent;
2. are not trivial, in the sense that the scenarios con-sidering all typical properties of the HEAD that can be consistently ascribed to C are discarded;
3. are those giving preference to the typical prop-erties of the HEAD CH (with respect to those
of the MODIFIER CM) with the highest
prob-ability. Notice that, in case of conflicting typi-cal properties like D and ¬D, given two scenar-ios w1and w2, both belonging to the set of
con-sistent scenarios with the highest probability and such that an inclusion p1 :: T(CH) v D
be-longs to w1 whereas p2 :: T(CM) v ¬D
be-longs to w2, the scenario w2is discarded in favor
of w1.
In order to select the desired scenarios we apply points 1, 2, and 3 above to blocks of scenarios with the same probability, in decreasing order starting from the high-est one. More in detail, we first discard all the incon-sistent scenarios, then we consider the remaining (con-sistent) ones in decreasing order by their probabilities. We then consider the blocks of scenarios with the same probability, and we proceed as follows:
– we discard those considered as trivial, consis-tently inheriting all (or most of) the typical prop-erties from the starting concepts to be combined; – among the remaining ones, we discard those in-heriting typical properties from the MODIFIER in conflict with typical properties inherited from the HEAD in another scenario of the same block (i.e., with the same probability);
– if the set of scenarios of the current block is empty, i.e. all the scenarios have been discarded either because trivial or because preferring the MODIFIER, we repeat the procedure by consid-ering the block of scenarios, all having the imme-diately lower probability.
The set of scenarios not discarded in the current block are those selected by the logic TCLas the result of the
procedure.
The knowledge base obtained as the result of com-bining concepts CHand CM into the compound
con-cept C is called C-revised knowledge base: KC = hR, T ∪ {p : T(C) v D}, Ai,
for all D such that T(C) v D belongs to w. The prob-ability p is defined as follows: if D is a typical prop-erty inherited either from the HEAD (or from both the HEAD and the MODIFIER), then p corresponds to the probability of such inclusion of the HEAD in the ini-tial knowledge base, i.e. p : T(CH) v D ∈ T ;
oth-erwise, p corresponds to the probability of such
inclu-sion of a MODIFIER in the initial knowledge base, i.e. p : T(CM) v D ∈ T . Notice that, since the
C-revisedknowledge base is still in the language of the TCL logic, we can iteratively repeat the same
proce-dure in order to combine not only atomic concepts, but also compound concepts. Examples and details of such procedure are described in [19].
In [18] we have shown that reasoning in TCLin the
revised knowledge is EXPTIME-complete, obtaining that it remains in the same complexity class of stan-dard ALC. For the completeness, let n be the size of KB, then the number of typicality inclusions is O(n). It is straightforward to observe that we have an expo-nential number of different scenarios, for each one we need to check whether the resulting KB is consistent in ALC + TRwhich is EXPTIME-complete. Hardness
immediately follows from the fact that TCL extends
ALC + TR.
It is worth noticing that, even if reasoning in the logic TCL remains in the same complexity class of
ALC, the generation of all 2nscenarios could be very
expensive in case of large knowledge bases. Optimiza-tion techniques like those in [1,2] could be used in or-der to limit this problem.
3. COCOS: a typicality based COncept COmbination System
In this section we describe COCOS [20], a first im-plementation of a reasoning machinery for the logic TCL, generating scenarios and choosing the selected
one(s) according to Section 2.1. The current version of the system is implemented in Python and exploits the translation of an ALC + TRknowledge base into
stan-dard ALC introduced in [7] and adopted by the sys-tem RAT-OWL [9]. COCOS makes use of the library owlready24that allows one to rely on the services of efficient DL reasoners, e.g. the HermiT reasoner.
Our system relies on the procedures developed for the logic TCLand recalled in the previous section.
CO-COS allows one: i) to include the logical descriptions of the concepts to be combined; ii) to select which among the concepts has to be intended as HEAD and as MODIFIER(s); iii) to choose, via a slider, how many typical properties we want to inherit in the scenarios that will be selected by TCL. In addition to
present-ing the selected scenario with typical properties of the
combined concept, COCOS also allows the users to se-lect alternative scenarios by means of a slider, ranging from more trivial to more surprising ones.
Some pictures of COCOS are shown in Figure 1. As mentioned, COCOS represents a very preliminary attempt to implement reasoning services for the logic TCL. We are currently working on a more mature
ver-sion by adopting techniques in [1,2] in order to im-prove its efficiency.
4. Artificial Prototypes Composition and Concept Invention
In this section we exploit the logic TCLto show both
how it allows to automatically generate novel, plau-sible, prototypical concepts by composing two initial prototypes and how it can be used as a generative tool in the field of computational creativity (with applica-tions in the so called creative industry). In detail, we first show how our logic can model the generation of a quite complex concept recently introduced in the field of narratology, i.e. that one of the Anti-Hero (a role in-vented by narratologists to generate new story lines), by combining the typical properties of the concepts Hero and Villain. Of course, the specific domain of the example is not relevant here; our goal is showing how TCL can model this kind of prototypical concept
composition (a crucial aspect of human concept inven-tion) that, on the other hand, has been proven to be problematic for other kinds of logics (e.g fuzzy logic, [11,23]). We then show how the same machinery can be used as a creative support tool to generate new char-acters, for instance for a video game or a movie. In particular, we first show how the logic TCL is able to
generate a concept like Batman as the combination of the prototypes of Man and Bat , then we exploit it to generate an entirely new fictional character that we as-sume to be composed by combining the narrative pro-totypes of Batman and of an Homeric Hero, as well as to create a new type of villain as the result of the combination with a chair.
For the examples we have exploited COCOS intro-duced in Section 3. The current version of the sys-tem along with the files for the following examples are available at http://di.unito.it/cocos. 4.1. The Anti Hero
We will take into account the concepts of Hero, AntiHero and Villain extracted by the common sense
descriptions coming from the TvTropes repository (https://tvtropes.org). In such online repos-itory, typical descriptions of character roles are pro-vided. They can be useful for practitioners of the nar-rative field in order to design their own character ac-cording to the main assets presented in such schemas. In particular, Tropes can be seen as devices and con-ventions that a writer can reasonably rely on as be-ing present in the audience members’ minds and ex-pectations. Regarding the Hero, TvTropes identifies the following relevant representative features: e.g. the fact that it is characterized by his/her fights against the Villain of a story, the fact that his/her actions are necessarily guided by general goals to be achieved in the interest of the collectivity, the fact that they fight against the Villain in a fair way and so on. Examples of such Trope are: Superman, Flash Gordon, Captain America, etc.. The AntiHero, on the other hand, is de-scribed as characterized by the fact of sharing most of its typical traits with the Hero (e.g. the fact that it is the protagonist of a plot fighting against the Villain of the story); however, his/her moves are not guided by a general spirit of sacrifice for the collectivity but, rather, they are usually based on some personal motivations that, incidentally and/or indirectly, coincide with the needs of the collectivity. Furthermore the AntiHero may also act in a not fair way in order to achieve the desired goal. A classical example of such trope is Bat-man, whose moves are guided by his desire of revenge. Finally the Villain is represented as a classic negative role in a plot and is characterized as the main opponent of the protagonist/Hero. In addition to this classical contraposition, TvTropes also reports some physical elements characterizing such role from a visual point of view. For example: the characters of this Trope are usually physically endowed with some demoniac cues (e.g. they have the “eyes of fire”). Finally, they are guided by negative moral values. Examples of such role can be easily taken from the classical literature to the modern comics. Some representative exemplars are Cruella de Vil in Disney’s filmic saga or Voldemort in Harry Potter.
Let us now exploit our logic TCLin order to define a
prototype of AntiHero. First of all, we define a knowl-edge base describing both rigid and typical properties of concepts Hero and Villain, then we rely on the logic TCL in order to formalize an AntiHero-revised
knowledge base.
Let K = hR, T , Ai be a KB, where the ABox A is empty. Concerning rigid properties, let R be as fol-lows:
Fig. 1. Some pictures of the graphical interface of COCOS.
Hero v ∃hasOpponent .Villain (R1)
Villain v ∀fightsFor .PersonalGoal (R2) Villain v WithNegativeMoralValues (R3) CollectiveGoal u PersonalGoal v ⊥ (R4) WithPositiveMoralValuesu
uWithNegativeMoralValues v ⊥ (R5) AngelicIconicity uDemoniacIconicity v ⊥ (R6) Prototypical properties of villains and heroes are de-scribed in T is as follows:
0.95 :: T(Hero) v Protagonist (T1)
0.85 :: T(Hero) v ∃fightsFor .CollectiveGoal (T2) 0.9 :: T(Hero) v WithPositiveMoralValues (T3)
0.6 :: T(Hero) v AngelicIconicity (T4) 0.75 :: T(Villain) v DemoniacIconicity (T5)
0.8 :: T(Villain) v Impulsive (T6)
0.75 :: T(Villain) v Protagonist (T7) We make use of the logic TCL in order to build the
compound concept AntiHero as the result of the com-bination of concepts Hero and Villain. Differently from what the natural language seems to suggest, we consider this compound concept by assuming that the HEAD is Villain (since the AntiHero shares more typical traits with this concept than with the Hero con-cept). In COCOS, such knowledg base is easily en-coded in a file containing:
– the choice about the HEAD concept and the MODIFIER one. This is provided by a key-value expression, in this case we just need to add the strings:
Head Concept Name: Villain Modifier Concept Name: Hero – the set of rigid and typical properties, expressed
by pairs head,concept_name modifier,concept_name and by triples T(head),concept_name,prob T(modifier),concept_name,prob respectively. In the example, for instance T1, R3 and T6 are encoded as follows:
T(modifier),protagonist,0.95 #T1 head,withnegativemoralvalues #R3 T(head),impulsive,0.8 #T6
First of all,e we have that the compound concepts in-herits all the rigid properties of both its components (if not contradictory), therefore in the logic TCL we have
that:
(i) AntiHero v ∃hasOpponent .Villain (ii) AntiHero v ∀fightsFor .PersonalGoal (iii) AntiHero v WithNegativeMoralValues
For the typical properties, we consider all the 27 =
256 different scenarios obtained from all possible se-lections about inclusion in T . Some of them are in-consistent, namely those including either axiom T2 or axiom T3, since they would ascribe typical properties in contrast with inherited rigid properties of (ii) and (iii): rigid properties impose that an anti hero has nega-tive moral values, and all his goals are personal, there-fore he is an atypical hero in those respects (T2 states that typical heroes fights also for some collective goals, whereas T3 states that normally heroes have positive moral values). Also scenarios containing both axioms T4 and T5 are inconsistent, due to the fact that the con-cepts AngelicIconicity and DemoniacIconicity are disjoint (formalized by R6).
Let us consider the remaining, consistent scenarios: the one having the highest probability considers all the typical properties of both concepts by excluding only AngelicIconicity, that is to say the one with the lowest probability between the two typical properties in conflict. In TCL this scenario is discarded since it
is the most trivial one. When we consider scenarios less trivial, i.e., more surprising scenarios (we analyze scenarios in decreasing order of probability), we dis-card the scenario with probability 0.13%, which
in-cludes T4, associated to the MODIFIER, rather than T5, associated to the HEAD, since it does not sat-isfy the HEAD/MODIFIER heuristics, allowing one to conclude, in a counter intuitive way, that typical anti heroes have an angelic iconicity rather than a demo-niac one.
Next scenarios, sharing the same probability (0.09%), are as follows: 0.95 :: T(Hero) v Protagonist (T1) 0.75 :: T(Villain) v DemoniacIconicity (T5) 0.8 :: T(Villain) v Impulsive (T6) 0.95 :: T(Hero) v Protagonist (T1) 0.8 :: T(Villain) v Impulsive (T6) 0.75 :: T(Villain) v Protagonist (T7) According to the logic TCL, both are adequate and
rep-resent the outcome of the whole heuristic procedures adopted in TCL. Probably, in this case, it could be more
useful to opt for the first solution allowing to inherit a further typical property (i.e. DemoniacIconicity) for the generated prototypical Anti-Hero. However, we re-main agnostic about the selection of the final options provided by TCL. This choice can be plausibly left to
human decision makers and based on their own goals. The resulting, alternative Hero u Villain-revised knowledge bases are as follows:
0.95 :: T(Hero u Villain) v Protagonist 0.75 :: T(HerouVillain) v DemoniacIconicity 0.8 :: T(Hero u Villain) v Impulsive
0.75 :: T(Hero u Villain) v Protagonist 0.8 :: T(Hero u Villain) v Impulsive
Notice that, in the second case, the inclusion T(Hero u Villain) v Protagonist is inherited from both the HEAD and the MODIFIER, therefore the probability in the revised knowledge base corresponds to the one coming from the HEAD (0.75).
COCOS ends its computation by providing the out-put of Figure 4.1, where tuples of 0 and 1 represent selections generating scenarios and the selected one is scenario numbered 19 at the end.
A final element that is worth noticing in TCLis the
following: in our logic, adding a new inclusion, e.g. T(AntiHero) v Brave, would not be problematic. That is to say that our formalism is able to tackle the phenomenon of prototypical attributes emergence for
the new compound concept, widely described in the Cognitive Science literature [10].
4.2. Batman
In this section we use the logic TCLto generate a
char-acter that is considered a typical Anti-Hero in the nar-ratological commununity: i.e. Batman. In this case, we assume that this character results from the combina-tion of the concepts Man and Bat , where Man is the HEAD and Bat is the MODIFIER.
First of all, let us consider the following list of pro-totypical descriptions for the concept Man (the list is, of course, plausible but not exhaustive):
Man v Male (R1) Man v Human (R2) Man v Mammal (R3) 0.7 :: T(Man) v HeteroSexual (T1) 0.85 :: T(Man) v SportLover (T2) 0.6 :: T(Man) v ¬AcuteVoice (T3) 0.8 :: T(Man) v Omnivore (T4)
Let us now formalize in the logic TCLthe prototype
of a bat, as follows: Bat v Mammal (R4) Bat v Animal (R5) 0.9 :: T(Bat ) v Fly (T5) 0.95 :: T(Bat ) v Nocturnal (T6) 0.9 :: T(Bat ) v ¬Omnivore (T7) 0.9 :: T(Bat ) v LivesInCave (T8)
We have exploited COCOS [20] in order to generate all 28 = 256 different scenarios. Excluding the incon-sistent scenarios, let us consider those remaining in de-scending order of probability. The first scenario, whose probability is 4.95%, includes typicality inclusions T1, T2, T3, T5, T6, T7, and T8, namely all typical proper-ties of the MODIFIER are inherited. However, this sce-nario is discarded since a typical property of the MOD-IFIER (¬Onnivore) is preferred to a conflict one in the HEAD (Onnivore). The same for subsequent scenario including T1, T2, T5, T6, T7, and T8, with probability 3.297%. The following scenario, excluding only T7, has probability 2.198%, but it is considered as trivial (all typicality properties of HEAD are inherited) and therefore discarded. The next scenario has probability 2.12% and includes T2, T3, T5, T6, T7, and T8. As for the most probable scenario, here again ¬Onnivore of T7 in the MODIFIER is preferred to the conflicting Onnivore of T4 in the HEAD: again, this scenario is therefore discarded.
...
([’0’, ’1’, ’1’, ’1’, ’0’, ’0’, ’0’, 0.08549999999999999], 19) * Owlready2 * Running HermiT...
...
... consisten scenario: OK ... ... NOT trivial scenario: OK ... ... HEAD/MODIFIER: OK ...
...
Result: Head Concept: Villain Modifier Concept: Hero
Recommended scenario(s) N.: 19
Fig. 2. An excerpt of the output of COCOS running the example of the AntiHero. In TCL the selected scenario, whose probability is
1.47% is the next one, including T1, T2, T4, T5, T6, and T8 and it describes the following Man u Bat -revised knowledge base:
0.7 :: T(Man u Bat ) v HeteroSexual (T1) 0.85 :: T(Man u Bat ) v SportLover (T2) 0.8 :: T(Man u Bat ) v Omnivore (T4)
0.9 :: T(Man u Bat ) v Fly (T5)
0.95 :: T(Man u Bat ) v Nocturnal (T6) 0.9 :: T(Man u Bat ) v LivesInCave (T8) This kind of revised knowledge base is perfectly com-pliant with that one generated in the previous section about the Anti-Hero. Also in this case, adding new in-clusions which are typical of an Anti-Hero such as:
0.95 :: T(Man u Bat ) v Protagonist (i) 0.8 :: T(Man u Bat ) v Impulsive (ii)
0.75 :: T(Man u Bat ) v Brave (iii)
would not be problematic. It is also true that, at this stage, the obtained description is also formally compli-ant with the both the descriptions of Hero and Villain. However, as soon as data about different instances of the new character class come in, e.g. by knowing that (Man u Bat )(michaelKeaton), we can assume that it will become evident that the following facts will hold:
(i) ∃hasOpponent .Villain(michaelKeaton) (ii) ∀fightsFor .PersonalGoal (michaelKeaton) (iii) WithNegativeMoralValues(michaelKeaton)
and therefore that correct narrative class of the novel generated character can only be the Anti-Hero one: in-deed, it can be assumed that Michael Keaton is a typi-cal Batman, therefore inheriting all the typitypi-cal proper-ties of the revised knowledge base. It can be observed that this does not hold in case we consider another in-dividual, call it Diomedes, being a Hero and a Batman:
in this case, the only consistent definition leads to as-sume that Diomedes in an atypical Batman.
In the next subsection we show how TCL can be
used to invent novel concepts by iteratively applying the heuristics devised in TCL.
4.3. Iterative Character Generation: Combining Batman with an Homeric Hero
A C-revised knowledge base in the logic TCLis still
in the language of the TCLlogic. This allows us to
iter-atively repeat the same procedure in order to combine not only atomic concepts, but also compound concepts. In this section we show how to generate a further new character by exploiting this feature, in particu-lar we handle the concept combination of C-revised knowledge bases in order to combine the concept Batman, obtained in Section 4.2 as the combination of Man and Bat , and the prototype of an HomericHero. For the sake of readability, let us recall the knowl-edge base concerning Batman:
Batman v Male (R1) Batman v Human (R2) Batman v Mammal (R3) Batman v Animal (R4) 0.7 :: T(Batman) v HeteroSexual (T1) 0.85 :: T(Batman) v SportLover (T2) 0.8 :: T(Batman) v Omnivore (T3) 0.9 :: T(Batman) v Fly (T5) 0.95 :: T(Batman) v Nocturnal (T4) 0.9 :: T(Batman) v LivesInCave (T6)
Concerning the Homeric hero, we can consider the fol-lowing knowledge base:
HomericHero v Brave (R5)
HomericHero v Valuable (6)
0.95 :: T(HomericHero) v Warrior (T8) 0.85 :: T(HomericHero) v
v ∃fightsFor .CollectiveGoals (T9) 0.8 :: T(HomericHero) v ¬Nocturnal (T10) We have considered the alternative of having BatMan as the HEAD and HomericHero as the MODIFIER and vice-versa and obtained the following revised knowledge bases. The Batman u HomericHero-revised knowledge base, where Batman is the HEAD, is as follows: Batman v Male (R1) Batman v Human (R2) Batman v Mammal (R3) Batman v Animal (R4) HomericHero v Brave (R5) HomericHero v Valuable (R6) 0.7 :: T(Batman) v HeteroSexual (T1) 0.85 :: T(Batman) v SportLover (T2) 0.8 :: T(Batman) v Omnivore (T3) 0.9 :: T(Batman) v Fly (T5) 0.95 :: T(Batman) v Nocturnal (T4) 0.9 :: T(Batman) v LivesInCave (T6) 0.9 :: T(HomericHero) v Male (T7) 0.95 :: T(HomericHero) v Warrior (T8) 0.85 :: T(HomericHero) v v ∃fightsFor .CollectiveGoals (T9) 0.8 :: T(HomericHero) v ¬Nocturnal (T10) 0.85 :: T(BatmanuHomericHero) v SportLover 0.8 :: T(Batman uHomericHero) v Omnivore 0.9 :: T(Batman u HomericHero) v Fly 0.95 :: T(BatmanuHomericHero) v Nocturnal 0.9 :: T(BatmanuHomericHero) v LivesInCave 0.9 :: T(Batman u HomericHero) v Male 0.95 :: T(Batman u HomericHero) v Warrior 0.85 :: T(Batman u HomericHero) v
v ∃fightsFor .CollectiveGoals It is worth noticing that in the revised knowledge base, the typical batman homeric hero is nocturnal, thus con-tradicting the typical property of one of the super-classes (i.e. HomericHero). This datum does not con-stitute a problem since the logic TCL is based on the
nonmonotonic logic ALC + TRwhich allows one to
handle this kind of exceptions by the nonmonotonic-ity of the operator T (C v D does not imply that T(C) v T(D), then we can have a property P holding for elements of T(C) whereas ¬P holds for elements of T(D)). Notice also that the combined concept Batman u HomericHero inherits the typical property of being male from the MODIFIER of the
combina-tion, i.e. 0.9 :: T(Batman u HomericHero) v Male from T7. This inclusion is allowed and it is not prob-lematic because it is not in conflict with the rigid prop-erty R1 stating that all batmans are males.
The HomericHero u Batman-revised knowledge base, where HomericHero is the HEAD, is as follows:
Batman v Male (R1) Batman v Human (R2) Batman v Mammal (R3) Batman v Animal (R4) HomericHero v Brave (R5) HomericHero v Valuable (R6) 0.7 :: T(Batman) v HeteroSexual (T1) 0.85 :: T(Batman) v SportLover (T2) 0.8 :: T(Batman) v Omnivore (T3) 0.9 :: T(Batman) v Fly (T5) 0.95 :: T(Batman) v Nocturnal (T4) 0.9 :: T(Batman) v LivesInCave (T6) 0.9 :: T(HomericHero) v Male (T7) 0.95 :: T(HomericHero) v Warrior (T8) 0.85 :: T(HomericHero) v v ∃fightsFor .CollectiveGoals (T9) 0.8 :: T(HomericHero) v ¬Nocturnal (T10) 0.7 :: T(HomericHerouBatman) v HeteroSexual 0.85 :: T(HomericHerouBatman) v SportLover 0.8 :: T(HomericHero uBatman) v Omnivore 0.9 :: T(HomericHero u Batman) v Fly
0.9 :: T(HomericHerouBatman) v LivesInCave 0.9 :: T(HomericHero u Batman) v Male 0.95 :: T(HomericHero u Batman) v Warrior 0.85 :: T(HomericHero u Batman) v
v ∃fightsFor .CollectiveGoals As a difference with the Batman u HomericHero-revised knowledge base, the typical property of be-ing Nocturnal is no longer inherited by the combined concept. It is worth noticing that, even if the typical property ¬Nocturnal would have been inherited, this would not be problematic, again by the nonmonotonic-ity of the underlying logic ALC + TR.
4.4. The Villain Chair
Let us assume to generate a novel concept obtained as the combination of concepts Villain (as HEAD)5
and Chair (as MODIFIER). Let K = hR, T , ∅i be as follows:
5Please, note that with respect to Section 4.1 we have slightly
Villain v ∃fightsFor .PersonalGoal (R1)
Villain v Animate (R2)
Villain v WithNegativeMoralValues (R3) Chair v ∃hasComponent .
.SupportingSeatComponent (R4)
Chair v ∃hasComponent .Seat (R5)
CollectiveGoal u PersonalGoal v ⊥ (R6) and T is as follows:
0.9 :: T(Villain) v DemoniacIconicity (T1) 0.75 :: T(Villain) v ∃hasOpponent .Hero (T2) 0.75 :: T(Villain) v Protagonist (T3)
0.8 :: T(Villain) v Impulsive (T4)
0.95 :: T(Chair ) v ¬Animate (T5)
0.95 :: T(Chair ) v ∃hasComponent .Back (T6) 0.65 :: T(Chair ) v ∃madeOf .Wood (T7)
0.8 :: T(Chair ) v Comfortable (T8)
0.7 :: T(Chair ) v Inflammable (T9)
We consider the 512 scenarios, from which we discard the inconsistent ones, namely those including T5: in-deed, since R2 imposes that villains are animate, in the underlying ALC + TR we conclude that Villain u
Chair v Animate, therefore all scenarios including T5, imposing that T(Villain u Chair ) v ¬Animate are consistent only if there are no villain chairs. As in the previous examples, we also discard trivial scenar-ios containing all the inclusions related to the HEAD. The first suitable scenarios are those have probability 0.23% and contain all typical properties coming from the MODIFIER and three out of four typical properties coming from the HEAD. Such scenarios define two al-ternative revised knowledge bases: one containing T2 and not T3, the other one containing T3 and not T2. These scenarios are the preferred ones selected by the logic TCL.
However, in this application setting, we could imag-ine to use our framework as a creativity support tool and thus considering alternative more surprising -scenarios by adding additional constraints. For ex-ample, we could impose that the compound concept should inherit exactly six typical properties. In this case, we would get that the scenario having the highest probability (0.16%) is the one including all the typical properties of the HEAD, namely T1, T2, T3 and T4, and two out of four typical properties of the MODI-FIER, namely T6 and T8. Due to its triviality, this sce-nario is discarded, in favor of the following more cre-ative scenarios, with probability 0.13%, obtained by excluding T7 of the MODIFIER and one out of four typical properties of the HEAD:
0.9 :: T(Villain u Chair ) v DemoniacIconicity 0.75 :: T(Villain u Chair ) v ∃hasOpponent .Hero 0.8 :: T(Villain u Chair ) v Impulsive
0.95 :: T(Villain u Chair ) v ∃hasComponent .Back 0.8 :: T(Villain u Chair ) v Comfortable
0.7 :: T(Villain u Chair ) v Inflammable 0.9 :: T(Villain u Chair ) v DemoniacIconicity 0.75 :: T(Villain u Chair ) v Protagonist 0.8 :: T(Villain u Chair ) v Impulsive
0.95 :: T(Villain u Chair ) v ∃hasComponent .Back 0.8 :: T(Villain u Chair ) v Comfortable
0.7 :: T(Villain u Chair ) v Inflammable
5. Related and Future Works
In this work we have considered a nonmonotonic Description Logic TCL, extending the DL of
typical-ity ALC + TRwith a DISPONTE semantics, in order
to deal with the generation of novel creative concepts. This logic enjoys good computational properties, since entailment in it remains ExpTime as the underlying monotonic ALC, and is able to take into account the concept combination of prototypical properties. To this aim, the logic TCLallows one to have inclusions of the
form p :: T(C) v D, representing that, with a prob-ability p, typical Cs are also Ds. Then, several differ-ent scenarios – having differdiffer-ent probabilities – are de-scribed by including or not such inclusions, and pro-totypical properties of combinations of concepts are obtained by restricting reasoning services to scenarios having suitable probabilities, excluding “trivial” ones with the highest probabilities.
In AI, several approaches have been proposed to deal with the problem of prototypical concept com-position in a human-like fashion. The authors of [14] present a detailed analysis of the limits of the set-theoretic approaches, the fuzzy logics (whose limi-tations was already shown in [28]), the vector-space models and quantum probability approaches proposed to model this phenomenon. In addition, they propose to use hierarchical conceptual spaces [5] to model the phenomenon in a way that accurately reflects how hu-mans exploit their creativity in conjunctive concept combination. While we agree with the authors with the comments moved to the described approaches, we showed that our logic can equally model, in a cogni-tively compliant-way, the composition of prototypes by using a nonmonotonic formalism whose
complex-ity remains in the same class of standard monotonic ALC. Other attempts similar to the one proposed here concerns the conceptual blending: a task where the ob-tained concept is entirely novel and has no strong as-sociation with the two base concepts (while in concept combination the compound concept is always a subset of the base concepts, for details see [22]). In [3] the authors propose a mechanism for conceptual blending based on the DL EL++. They construct the generic space of two concepts by introducing an upward re-finement operator that is used for finding common gen-eralizations of EL++ concepts. However, differently from us, what they call prototypes are expressed in the standard monotonic DL, which does not allow one to reason about typicality and defeasible inheritance. More recently, a different approach is proposed in [4], where the authors see the problem of concept blend-ing as a nonmonotonic search problem and propose to use Answer Set Programming (ASP) to deal with this search problem in a nonmonotonic way. In a related work [25], the author extends the logic of typicality ALC + TRby means of probabilities equipping
typ-icality inclusions of the form T(C) vp D, whose
in-tuitive meaning is that, “normally, Cs are Ds and we have a probability of 1−p of having exceptional Cs not being Ds”. Probabilities of exceptions are then used in order to reason about plausible scenarios, obtained by selecting only some typicality assumptions and whose probabilities belong to a given and fixed range. As a difference with the logic TCL, all typicality
assump-tions are systematically taken into account: as a con-sequence, one cannot exploit such a DL for capturing compositionality, since it is not possible to block in-heritance of prototypical properties in concept combi-nation. The logic TCLextends the work of [25] in that
it does not systematically take into account all typical-ity assumptions. As a consequence, TCLblocks
inher-itance of prototypical properties in concept combina-tion. The same criticism applies also to the approach proposed in [24], where ALC + TR is extended by
inclusions of the form T(C) vd D, where d is a
de-gree of expectedness, used to define a preference rela-tion among extended ABoxes: entailment of queries is then restricted to ABoxes that are minimal with respect to such preference relations and that represent surpris-ing scenarios. Also in this case, however, the result-ing logic does not allow us to define scenarios contain-ing only some inclusions, since all of them are system-atically considered. Similarly, probabilistic DLs [26] themselves cannot be employed as a framework for dealing with the combination of concepts, since these
logics are not able to represent and reason about pro-totypical properties.
In future research we aim at extending our approach to more expressive DLs, such as those underlying the standard OWL language. Starting from the work of [6], applying the logic with the typicality operator and the rational closure to SHIQ, we intend to study whether and how TCL could provide an alternative solution to
the problem of the “all or nothing” behavior of ratio-nal closure with respect to property inheritance. Other applications field of the proposed approach other em-ployments can be considered. For example, we plan to use our logic for the development of a goal-directed and automatic generation of novel knowledge in a cog-nitive artificial agent, starting from an initial common-sense knowledge base. This approach will require to enrich the knowledge processing mechanisms of cur-rent cognitive agents [16] since it does not assume that the only way to process and reason on new knowledge is via an external knowledge injection. On the con-trary, it assumes that a mechanism for the automatic and creative re-framing, or re-formulation, of the avail-able knowledge. Such procedure, in particular, can be used to extend the mechanisms of universal subgoal-ingin cognitive architectures like SOAR [12].
In this work we have also described COCOS, a tool for combining concepts in the logic TCL. As
men-tioned, COCOS represents a very preliminary attempt to implement reasoning services for the logic TCL, and
a more mature version, exploiting optimization tech-niques in [1,2] in order to improve its efficiency, is cur-rently under development.
Acknowledgements
This work has been partially supported by the project “METALLIC #2: METodi di prova per il ra-gionamento Automatico per Logiche non-cLassIChe #2”, Progetto di Ricerca INdAM GNCS 2019.
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